The invention relates to a method and device for determining the intensity and/or phase distribution in various cut planes of a space coherent beam, in particular of the light beam produced by a laser.
To carry out such a determination, known devices are split into three categories:
1) interferometry devices,
2) devices of Hartmann-Shack type
3) space pattern sampling devices.
Interferometry devices and methods use the combination of the wave front to be measured with a wave having a phase relation therewith.
Devices and methods of the Hartmann-Shack type use a phase mask (lens arrays, holes . . . ) to evaluate the slope of the wave surface at each point. Reconstruction then allows the wave surface to be calculated.
The problem of obtaining the phase from mere intensity profiles has already been studied in literature as the so-called xe2x80x9cphase retrieval problemxe2x80x9d. The study of various existing solutions is proposed in the articles xe2x80x9cPhase retrieval algorithms: a comparisonxe2x80x9d, Applied Optics, vol. 21, p. 2758, August 1982, and xe2x80x9cThe phase retrieval problemxe2x80x9d, IEEE Transactions on Antennas and Propagation AP29, P. 386, March 1981.
In particular, the algorithm proposed by Gerchberg and Saxton in 1971 (magazine Optik 34, 275 (1971)) is known. The authors use an image plane, and a distant field diffraction plane, at the focus of a lens.
According to this algorithm, an intensity profile is measured in a first cut plane, by assigning a phase profile that is arbitrary in every point of this plane; next, using beam propagation equations, the intensity and phase profile in a second cut plane is calculated: the intensity profile calculated in this second plane is replaced by the measured profile, and the intensity and phase profiles are again calculated in the first plane. Iteration is performed up to convergence.
According to the article by D. L. Misell (xe2x80x9cA method for the solution of the phase problem in electron microscopyxe2x80x9d) published in J. Phys. D. Appl. Phys., vol. 6, 1973, p. L6-L9 and the comments on this article by Gerchberg and Saxton (same references, p. L31), both planes can be any planes but close to each other.
In recent literature, numerous studies have been conducted on these methods, applied to electronic microscopy and antenna transmission. These studies suggest algorithmic improvements. E.g., we may quote the article xe2x80x9cRadiation Pattern Evaluation from near-field Intensities on planesxe2x80x9d, IEEE Transactions on Antennas and Propagation, vol. 44, no. 5, May 1996, which suggests not to use necessarily the object plane and the Fourier plane.
Tests performed with laser beam images and these methods have not allowed to reach results sufficiently good to make it possible to do without systems of the Hartmann-Shack type.
The problem is therefore posed of finding a method and device allowing to improve the accuracy of phase reconstruction, and compete with expensive commercially available systems, e.g. of Hartmann-Shack or Zygo type.
First of all, the object of the invention is an iterative method of determining the intensity and phase distribution of a light beam coherent in one plane, comprising the steps of:
measuring the intensity Ii, i=1 . . . N, of the beam, in N planes, Nxe2x89xa73, including the plane wherein intensity and phase distribution is to be determined,
choosing, for plane i=1, an initial phase matrix xcfx861 and calculating a complex amplitude matrix, by term-wise multiplying the phase matrix eixcfx861 by the corresponding amplitude matrix A1,
for each plane j greater than 1:
determining a propagated complex matrix Bxe2x80x2j from the measured amplitude matrix Ijxe2x88x921 of plane jxe2x88x921 and the phase matrix xcfx86 of plane jxe2x88x921,
extracting, from Bxe2x80x2j, the phase matrix of plane j,
iterating the method up to convergence (and j=1 when jxe2x88x921=N).
The method according to the invention therefore implements N planes, with Nxe2x89xa73. One of these planes is that of which intensity and phase distribution is to be determined. The method starts with a phase profile (e.g., null) on a first plane. Then, by calculation, the beam propagation from one plane to another are performed while keeping, after each iteration, the phase profile calculated for a plane, and by introducing the intensity profile measured in the same cut plane. Thus, we proceed from plane #1 to plane #2, . . . to plane #N, then we return to plane #1. The convergence of the phase profile towards the intended profile is noted: indeed, the intensity and phase profiles calculated in one of the N planes allow the intensity profiles measured in any other plane to be retrieved exactly.
For each plane j(j greater than 1), a phase profile is calculated from the complex matrix reflecting the state of the beam in this plane. This complex matrix is itself the result of a calculation, which reflects the equations of propagation, implementing the (measured) intensity matrix and the (calculated) phase matrix in plane jxe2x88x921. In other words, for each plane j, by propagation, the intensity and phase profile in this plane is calculated, then the intensity profile calculated in this plane is replaced by the measured intensity profile. It is then possible, from this measured intensity profile and the calculated phase profile, to perform a propagation calculation for plane j+1.
It is possible to use any path among the N planes, as required.
Consequently, according to the invention, instead of moving to and fro among two cut planes, at least three cut planes are implemented and we proceed for instance from the first plane to the second one, then to the third one, . . . then to the Nth one, and we return to the first plane, and so on. The accuracy of the phase reconstruction on the intermediate planes is thus improved significantly, which allows to compete with expensive commercially available systems, e.g. of Hartmann-Shack or Zygo type.
According to a development, the phase taken into consideration in the first cut plane at each iteration is an average of the phase profiles determined from all the plane pairs in one round trip. This method allows to introduce a weighting factor into averaging, wherein this factor can depend on a merit function of the current calculation. Thus, convergence on the N profiles is optimized.
Two versions of the method according to the invention are therefore proposed according to the calculation path among the measuring planes. The first methods covers the planes sequentially, whereas the second one performs a round-trip among plane pairs, and allows the relative weight of the different measured planes to be weighted.
The invention also relates to a device for implementing the invention.
Thus, the invention relates to a device, for determining the intensity and phase of a light beam in a plane, comprising:
a means for measuring intensity Ii (i=1, . . . N) of the beam in N planes, Nxe2x89xa73,
based on an initial phase matrix xcfx861 corresponding to a plane i=1, a calculation means programmed for:
calculating a complex amplitude matrix (B1) by multiplying the phase matrix eixcfx861 by the corresponding amplitude matrix (A1),
for each plane j greater than 1:
calculating a propagated complex matrix Bxe2x80x2j from the measured amplitude matrix Ajxe2x88x921 of plane jxe2x88x921 and the phase matrix xcfx86 of plane jxe2x88x921,
extracting a phase matrix from plane j of matrix Bxe2x80x2j,
iterating the above calculation by making j=1 when jxe2x88x921=N.
This device uses as many cameras as there are cut planes; preferably, the measured images are digitized before being processed. It can include a system for shaping the beam, limiting the extend thereof so as to adapt it to the cameras and to the bank length; a calibration step is then integrated in the steps of the main method. The addition of a lens or any other phase object allowing the beam to be adapted to the measuring device (divergence control . . . ) makes it possible to increase the field of application of the device. Using calculations, the calibration step then allows to do without the presence of the added component and thus to gain access to beam characteristics.
A phase analyzer using a method according to the invention allows to obtain the complex amplitude distribution of the field, representing both beam intensity and phase in a given plane (generally an image plane). This measurement then allows the beam progress in other planes to be calculated directly and reliably.